<p>The number of different ways to divide a set P of \(n\) elements into two nonempty disjoint subsets whose union is P, is</p>
Step-by-Step Solution
Key Concept: Each element can go to either subset A or B (2^n ways), but we must exclude empty subsets and account for the fact that {A,B} and {B,A} represent the same partition (unordered division).
<p><strong>Step 1:</strong> Each of the n elements can independently go into subset A or subset B, giving 2^n total assignments.</p><p><strong>Step 2:</strong> Exclude cases where all elements go to A (1 way) or all go to B (1 way), leaving 2^n - 2 ways for nonempty subsets.</p><p><strong>Step 3:</strong> Since we're dividing P into two nonempty <em>unordered</em> disjoint subsets, the partition {A, B} is identical to {B, A}. We've counted each partition twice.</p><p><strong>Step 4:</strong> Divide by 2 to account for this over-counting: (2^n - 2)/2 = <strong>2^(n-1) - 1</strong></p><p>∴ Answer: A</p>
Correct Answer: A